LMFDB - Embedded newform 6030.2.d.j.2411.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 519x^{12} + 3876x^{10} + 16111x^{8} + 36772x^{6} + 41293x^{4} + 16036x^{2} + 4 $ x^16 + 36x^14 + 519x^12 + 3876x^10 + 16111x^8 + 36772x^6 + 41293x^4 + 16036*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{67}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.13
Root			$2.22811i$ of defining polynomial
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.j.2411.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.75929i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.75929i q^{7} +1.00000 q^{8} -1.00000 q^{10} -0.476212 q^{11} -4.74819i q^{13} +3.75929i q^{14} +1.00000 q^{16} -4.08432i q^{17} -3.12553 q^{19} -1.00000 q^{20} -0.476212 q^{22} +0.982457i q^{23} +1.00000 q^{25} -4.74819i q^{26} +3.75929i q^{28} -5.08310i q^{29} -1.51012i q^{31} +1.00000 q^{32} -4.08432i q^{34} -3.75929i q^{35} -4.80373 q^{37} -3.12553 q^{38} -1.00000 q^{40} -8.34118 q^{41} -1.30716i q^{43} -0.476212 q^{44} +0.982457i q^{46} +0.0921822i q^{47} -7.13228 q^{49} +1.00000 q^{50} -4.74819i q^{52} +1.41730 q^{53} +0.476212 q^{55} +3.75929i q^{56} -5.08310i q^{58} +11.8782i q^{59} +12.8767i q^{61} -1.51012i q^{62} +1.00000 q^{64} +4.74819i q^{65} +(-7.79027 + 2.51230i) q^{67} -4.08432i q^{68} -3.75929i q^{70} -13.0140i q^{71} -7.77609 q^{73} -4.80373 q^{74} -3.12553 q^{76} -1.79022i q^{77} -9.74967i q^{79} -1.00000 q^{80} -8.34118 q^{82} -15.7344i q^{83} +4.08432i q^{85} -1.30716i q^{86} -0.476212 q^{88} -15.4861i q^{89} +17.8498 q^{91} +0.982457i q^{92} +0.0921822i q^{94} +3.12553 q^{95} -6.02861i q^{97} -7.13228 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			3.75929i			1.42088i	0.703758	+	0.710439i	$0.251504\pi$
$7$			3.75929i			1.42088i	−0.703758	+	0.710439i	$0.748496\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−0.476212			−0.143583			−0.0717917	−	0.997420i	$-0.522872\pi$
$11$	−0.476212			−0.143583			−0.0717917	+	0.997420i	$0.522872\pi$
$12$		0			0
$13$		−	4.74819i		−	1.31691i	−0.752620	−	0.658456i	$-0.771210\pi$
$13$		−	4.74819i		−	1.31691i	0.752620	−	0.658456i	$-0.228790\pi$
$14$			3.75929i			1.00471i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	4.08432i		−	0.990592i	−0.868724	−	0.495296i	$-0.835060\pi$
$17$		−	4.08432i		−	0.990592i	0.868724	−	0.495296i	$-0.164940\pi$
$18$		0			0
$19$	−3.12553			−0.717046			−0.358523	−	0.933521i	$-0.616720\pi$
$19$	−3.12553			−0.717046			−0.358523	+	0.933521i	$0.616720\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	−0.476212			−0.101529
$23$			0.982457i			0.204856i	0.994740	+	0.102428i	$0.0326612\pi$
$23$			0.982457i			0.204856i	−0.994740	+	0.102428i	$0.967339\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	4.74819i		−	0.931197i
$27$		0			0
$28$			3.75929i			0.710439i
$29$		−	5.08310i		−	0.943909i	−0.881623	−	0.471954i	$-0.843549\pi$
$29$		−	5.08310i		−	0.943909i	0.881623	−	0.471954i	$-0.156451\pi$
$30$		0			0
$31$		−	1.51012i		−	0.271225i	−0.990762	−	0.135612i	$-0.956700\pi$
$31$		−	1.51012i		−	0.271225i	0.990762	−	0.135612i	$-0.0433002\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	4.08432i		−	0.700454i
$35$		−	3.75929i		−	0.635436i
$36$		0			0
$37$	−4.80373			−0.789728			−0.394864	−	0.918740i	$-0.629208\pi$
$37$	−4.80373			−0.789728			−0.394864	+	0.918740i	$0.629208\pi$
$38$	−3.12553			−0.507028
$39$		0			0
$40$	−1.00000			−0.158114
$41$	−8.34118			−1.30267			−0.651337	−	0.758789i	$-0.725791\pi$
$41$	−8.34118			−1.30267			−0.651337	+	0.758789i	$0.725791\pi$
$42$		0			0
$43$		−	1.30716i		−	0.199340i	−0.995021	−	0.0996700i	$-0.968221\pi$
$43$		−	1.30716i		−	0.199340i	0.995021	−	0.0996700i	$-0.0317787\pi$
$44$	−0.476212			−0.0717917
$45$		0			0
$46$			0.982457i			0.144855i
$47$			0.0921822i			0.0134462i	0.999977	+	0.00672308i	$0.00214004\pi$
$47$			0.0921822i			0.0134462i	−0.999977	+	0.00672308i	$0.997860\pi$
$48$		0			0
$49$	−7.13228			−1.01890
$50$	1.00000			0.141421
$51$		0			0
$52$		−	4.74819i		−	0.658456i
$53$	1.41730			0.194681			0.0973403	−	0.995251i	$-0.468966\pi$
$53$	1.41730			0.194681			0.0973403	+	0.995251i	$0.468966\pi$
$54$		0			0
$55$	0.476212			0.0642124
$56$			3.75929i			0.502357i
$57$		0			0
$58$		−	5.08310i		−	0.667444i
$59$			11.8782i			1.54641i	0.634158	+	0.773204i	$0.281347\pi$
$59$			11.8782i			1.54641i	−0.634158	+	0.773204i	$0.718653\pi$
$60$		0			0
$61$			12.8767i			1.64869i	0.566089	+	0.824344i	$0.308456\pi$
$61$			12.8767i			1.64869i	−0.566089	+	0.824344i	$0.691544\pi$
$62$		−	1.51012i		−	0.191785i
$63$		0			0
$64$	1.00000			0.125000
$65$			4.74819i			0.588941i
$66$		0			0
$67$	−7.79027	+	2.51230i	−0.951733	+	0.306926i
$67$	−7.79027	+	2.51230i	−0.951733	+	0.306926i
$68$		−	4.08432i		−	0.495296i
$69$		0			0
$70$		−	3.75929i		−	0.449321i
$71$		−	13.0140i		−	1.54448i	−0.635332	−	0.772239i	$-0.719137\pi$
$71$		−	13.0140i		−	1.54448i	0.635332	−	0.772239i	$-0.280863\pi$
$72$		0			0
$73$	−7.77609			−0.910123			−0.455062	−	0.890460i	$-0.650383\pi$
$73$	−7.77609			−0.910123			−0.455062	+	0.890460i	$0.650383\pi$
$74$	−4.80373			−0.558422
$75$		0			0
$76$	−3.12553			−0.358523
$77$		−	1.79022i		−	0.204015i
$78$		0			0
$79$		−	9.74967i		−	1.09692i	−0.836176	−	0.548462i	$-0.815214\pi$
$79$		−	9.74967i		−	1.09692i	0.836176	−	0.548462i	$-0.184786\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	−8.34118			−0.921129
$83$		−	15.7344i		−	1.72707i	−0.504285	−	0.863537i	$-0.668244\pi$
$83$		−	15.7344i		−	1.72707i	0.504285	−	0.863537i	$-0.331756\pi$
$84$		0			0
$85$			4.08432i			0.443006i
$86$		−	1.30716i		−	0.140955i
$87$		0			0
$88$	−0.476212			−0.0507644
$89$		−	15.4861i		−	1.64153i	−0.571268	−	0.820764i	$-0.693548\pi$
$89$		−	15.4861i		−	1.64153i	0.571268	−	0.820764i	$-0.306452\pi$
$90$		0			0
$91$	17.8498			1.87117
$92$			0.982457i			0.102428i
$93$		0			0
$94$			0.0921822i			0.00950787i
$95$	3.12553			0.320673
$96$		0			0
$97$		−	6.02861i		−	0.612113i	−0.952013	−	0.306056i	$-0.900990\pi$
$97$		−	6.02861i		−	0.612113i	0.952013	−	0.306056i	$-0.0990096\pi$
$98$	−7.13228			−0.720469
$99$		0			0
$100$	1.00000			0.100000
$101$	1.15912			0.115337			0.0576685	−	0.998336i	$-0.481633\pi$
$101$	1.15912			0.115337			0.0576685	+	0.998336i	$0.481633\pi$
$102$		0			0
$103$	16.4429			1.62017			0.810083	−	0.586315i	$-0.199422\pi$
$103$	16.4429			1.62017			0.810083	+	0.586315i	$0.199422\pi$
$104$		−	4.74819i		−	0.465598i
$105$		0			0
$106$	1.41730			0.137660
$107$		−	11.5125i		−	1.11295i	−0.830863	−	0.556476i	$-0.812153\pi$
$107$		−	11.5125i		−	1.11295i	0.830863	−	0.556476i	$-0.187847\pi$
$108$		0			0
$109$			6.66875i			0.638750i	0.947628	+	0.319375i	$0.103473\pi$
$109$			6.66875i			0.638750i	−0.947628	+	0.319375i	$0.896527\pi$
$110$	0.476212			0.0454050
$111$		0			0
$112$			3.75929i			0.355220i
$113$	11.9910			1.12802			0.564009	−	0.825768i	$-0.309258\pi$
$113$	11.9910			1.12802			0.564009	+	0.825768i	$0.309258\pi$
$114$		0			0
$115$		−	0.982457i		−	0.0916146i
$116$		−	5.08310i		−	0.471954i
$117$		0			0
$118$			11.8782i			1.09348i
$119$	15.3541			1.40751
$120$		0			0
$121$	−10.7732			−0.979384
$122$			12.8767i			1.16580i
$123$		0			0
$124$		−	1.51012i		−	0.135612i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−1.66564			−0.147802			−0.0739008	−	0.997266i	$-0.523545\pi$
$127$	−1.66564			−0.147802			−0.0739008	+	0.997266i	$0.523545\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			4.74819i			0.416444i
$131$		−	19.5370i		−	1.70695i	−0.521133	−	0.853476i	$-0.674490\pi$
$131$		−	19.5370i		−	1.70695i	0.521133	−	0.853476i	$-0.325510\pi$
$132$		0			0
$133$		−	11.7498i		−	1.01884i
$134$	−7.79027	+	2.51230i	−0.672977	+	0.217029i
$135$		0			0
$136$		−	4.08432i		−	0.350227i
$137$	−9.32667			−0.796831			−0.398415	−	0.917205i	$-0.630440\pi$
$137$	−9.32667			−0.796831			−0.398415	+	0.917205i	$0.630440\pi$
$138$		0			0
$139$		−	19.2238i		−	1.63054i	−0.579080	−	0.815271i	$-0.696588\pi$
$139$		−	19.2238i		−	1.63054i	0.579080	−	0.815271i	$-0.303412\pi$
$140$		−	3.75929i		−	0.317718i
$141$		0			0
$142$		−	13.0140i		−	1.09211i
$143$			2.26115i			0.189086i
$144$		0			0
$145$			5.08310i			0.422129i
$146$	−7.77609			−0.643554
$147$		0			0
$148$	−4.80373			−0.394864
$149$			7.15655i			0.586288i	0.956068	+	0.293144i	$0.0947015\pi$
$149$			7.15655i			0.586288i	−0.956068	+	0.293144i	$0.905299\pi$
$150$		0			0
$151$	12.9766			1.05602			0.528010	−	0.849238i	$-0.322938\pi$
$151$	12.9766			1.05602			0.528010	+	0.849238i	$0.322938\pi$
$152$	−3.12553			−0.253514
$153$		0			0
$154$		−	1.79022i		−	0.144260i
$155$			1.51012i			0.121295i
$156$		0			0
$157$	−18.2074			−1.45310			−0.726552	−	0.687111i	$-0.758879\pi$
$157$	−18.2074			−1.45310			−0.726552	+	0.687111i	$0.758879\pi$
$158$		−	9.74967i		−	0.775642i
$159$		0			0
$160$	−1.00000			−0.0790569
$161$	−3.69334			−0.291076
$162$		0			0
$163$	−2.72596			−0.213513			−0.106757	−	0.994285i	$-0.534047\pi$
$163$	−2.72596			−0.213513			−0.106757	+	0.994285i	$0.534047\pi$
$164$	−8.34118			−0.651337
$165$		0			0
$166$		−	15.7344i		−	1.22123i
$167$		−	1.15786i		−	0.0895982i	−0.998996	−	0.0447991i	$-0.985735\pi$
$167$		−	1.15786i		−	0.0895982i	0.998996	−	0.0447991i	$-0.0142648\pi$
$168$		0			0
$169$	−9.54531			−0.734255
$170$			4.08432i			0.313253i
$171$		0			0
$172$		−	1.30716i		−	0.0996700i
$173$		−	6.09577i		−	0.463453i	−0.972781	−	0.231726i	$-0.925563\pi$
$173$		−	6.09577i		−	0.463453i	0.972781	−	0.231726i	$-0.0744374\pi$
$174$		0			0
$175$			3.75929i			0.284176i
$176$	−0.476212			−0.0358958
$177$		0			0
$178$		−	15.4861i		−	1.16074i
$179$	−4.62351			−0.345577			−0.172789	−	0.984959i	$-0.555278\pi$
$179$	−4.62351			−0.345577			−0.172789	+	0.984959i	$0.555278\pi$
$180$		0			0
$181$	−2.93753			−0.218345			−0.109172	−	0.994023i	$-0.534820\pi$
$181$	−2.93753			−0.218345			−0.109172	+	0.994023i	$0.534820\pi$
$182$	17.8498			1.32312
$183$		0			0
$184$			0.982457i			0.0724277i
$185$	4.80373			0.353177
$186$		0			0
$187$			1.94500i			0.142233i
$188$			0.0921822i			0.00672308i
$189$		0			0
$190$	3.12553			0.226750
$191$	15.7717			1.14120			0.570601	−	0.821227i	$-0.306710\pi$
$191$	15.7717			1.14120			0.570601	+	0.821227i	$0.306710\pi$
$192$		0			0
$193$	−16.3858			−1.17948			−0.589739	−	0.807594i	$-0.700769\pi$
$193$	−16.3858			−1.17948			−0.589739	+	0.807594i	$0.700769\pi$
$194$		−	6.02861i		−	0.432829i
$195$		0			0
$196$	−7.13228			−0.509448
$197$	22.0080			1.56801			0.784004	−	0.620755i	$-0.213174\pi$
$197$	22.0080			1.56801			0.784004	+	0.620755i	$0.213174\pi$
$198$		0			0
$199$	−23.6827			−1.67882			−0.839411	−	0.543497i	$-0.817100\pi$
$199$	−23.6827			−1.67882			−0.839411	+	0.543497i	$0.817100\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	1.15912			0.0815555
$203$	19.1089			1.34118
$204$		0			0
$205$	8.34118			0.582573
$206$	16.4429			1.14563
$207$		0			0
$208$		−	4.74819i		−	0.329228i
$209$	1.48842			0.102956
$210$		0			0
$211$	24.0556			1.65605			0.828026	−	0.560689i	$-0.189464\pi$
$211$	24.0556			1.65605			0.828026	+	0.560689i	$0.189464\pi$
$212$	1.41730			0.0973403
$213$		0			0
$214$		−	11.5125i		−	0.786976i
$215$			1.30716i			0.0891476i
$216$		0			0
$217$	5.67697			0.385378
$218$			6.66875i			0.451665i
$219$		0			0
$220$	0.476212			0.0321062
$221$	−19.3931			−1.30452
$222$		0			0
$223$	4.78479			0.320413			0.160207	−	0.987083i	$-0.448784\pi$
$223$	4.78479			0.320413			0.160207	+	0.987083i	$0.448784\pi$
$224$			3.75929i			0.251178i
$225$		0			0
$226$	11.9910			0.797630
$227$		−	2.84112i		−	0.188572i	−0.995545	−	0.0942859i	$-0.969943\pi$
$227$		−	2.84112i		−	0.188572i	0.995545	−	0.0942859i	$-0.0300568\pi$
$228$		0			0
$229$		−	19.1224i		−	1.26364i	−0.775114	−	0.631821i	$-0.782308\pi$
$229$		−	19.1224i		−	1.26364i	0.775114	−	0.631821i	$-0.217692\pi$
$230$		−	0.982457i		−	0.0647813i
$231$		0			0
$232$		−	5.08310i		−	0.333722i
$233$	27.4779			1.80014			0.900069	−	0.435747i	$-0.143516\pi$
$233$	27.4779			1.80014			0.900069	+	0.435747i	$0.143516\pi$
$234$		0			0
$235$		−	0.0921822i		−	0.00601330i
$236$			11.8782i			0.773204i
$237$		0			0
$238$	15.3541			0.995261
$239$	−18.7228			−1.21108			−0.605538	−	0.795817i	$-0.707042\pi$
$239$	−18.7228			−1.21108			−0.605538	+	0.795817i	$0.707042\pi$
$240$		0			0
$241$	−1.66807			−0.107450			−0.0537250	−	0.998556i	$-0.517109\pi$
$241$	−1.66807			−0.107450			−0.0537250	+	0.998556i	$0.517109\pi$
$242$	−10.7732			−0.692529
$243$		0			0
$244$			12.8767i			0.824344i
$245$	7.13228			0.455664
$246$		0			0
$247$			14.8406i			0.944286i
$248$		−	1.51012i		−	0.0958925i
$249$		0			0
$250$	−1.00000			−0.0632456
$251$	−1.33716			−0.0844007			−0.0422004	−	0.999109i	$-0.513437\pi$
$251$	−1.33716			−0.0844007			−0.0422004	+	0.999109i	$0.513437\pi$
$252$		0			0
$253$		−	0.467858i		−	0.0294140i
$254$	−1.66564			−0.104512
$255$		0			0
$256$	1.00000			0.0625000
$257$			21.0269i			1.31162i	0.754926	+	0.655810i	$0.227673\pi$
$257$			21.0269i			1.31162i	−0.754926	+	0.655810i	$0.772327\pi$
$258$		0			0
$259$		−	18.0586i		−	1.12211i
$260$			4.74819i			0.294470i
$261$		0			0
$262$		−	19.5370i		−	1.20700i
$263$		−	12.8023i		−	0.789423i	−0.918805	−	0.394711i	$-0.870845\pi$
$263$		−	12.8023i		−	0.789423i	0.918805	−	0.394711i	$-0.129155\pi$
$264$		0			0
$265$	−1.41730			−0.0870639
$266$		−	11.7498i		−	0.720426i
$267$		0			0
$268$	−7.79027	+	2.51230i	−0.475867	+	0.153463i
$269$			10.8084i			0.658999i	0.944156	+	0.329500i	$0.106880\pi$
$269$			10.8084i			0.658999i	−0.944156	+	0.329500i	$0.893120\pi$
$270$		0			0
$271$			8.07489i			0.490514i	0.969458	+	0.245257i	$0.0788724\pi$
$271$			8.07489i			0.490514i	−0.969458	+	0.245257i	$0.921128\pi$
$272$		−	4.08432i		−	0.247648i
$273$		0			0
$274$	−9.32667			−0.563445
$275$	−0.476212			−0.0287167
$276$		0			0
$277$	−27.0641			−1.62612			−0.813062	−	0.582177i	$-0.802201\pi$
$277$	−27.0641			−1.62612			−0.813062	+	0.582177i	$0.802201\pi$
$278$		−	19.2238i		−	1.15297i
$279$		0			0
$280$		−	3.75929i		−	0.224661i
$281$	−4.54839			−0.271334			−0.135667	−	0.990754i	$-0.543318\pi$
$281$	−4.54839			−0.271334			−0.135667	+	0.990754i	$0.543318\pi$
$282$		0			0
$283$	−6.53037			−0.388190			−0.194095	−	0.980983i	$-0.562177\pi$
$283$	−6.53037			−0.388190			−0.194095	+	0.980983i	$0.562177\pi$
$284$		−	13.0140i		−	0.772239i
$285$		0			0
$286$			2.26115i			0.133704i
$287$		−	31.3569i		−	1.85094i
$288$		0			0
$289$	0.318365			0.0187273
$290$			5.08310i			0.298490i
$291$		0			0
$292$	−7.77609			−0.455062
$293$			20.0046i			1.16868i	0.811508	+	0.584342i	$0.198647\pi$
$293$			20.0046i			1.16868i	−0.811508	+	0.584342i	$0.801353\pi$
$294$		0			0
$295$		−	11.8782i		−	0.691574i
$296$	−4.80373			−0.279211
$297$		0			0
$298$			7.15655i			0.414568i
$299$	4.66489			0.269778
$300$		0			0
$301$	4.91400			0.283238
$302$	12.9766			0.746719
$303$		0			0
$304$	−3.12553			−0.179262
$305$		−	12.8767i		−	0.737316i
$306$		0			0
$307$	6.10978			0.348703			0.174352	−	0.984683i	$-0.444217\pi$
$307$	6.10978			0.348703			0.174352	+	0.984683i	$0.444217\pi$
$308$		−	1.79022i		−	0.102007i
$309$		0			0
$310$			1.51012i			0.0857688i
$311$	−4.02204			−0.228069			−0.114034	−	0.993477i	$-0.536377\pi$
$311$	−4.02204			−0.228069			−0.114034	+	0.993477i	$0.536377\pi$
$312$		0			0
$313$		−	15.8028i		−	0.893229i	−0.894727	−	0.446614i	$-0.852630\pi$
$313$		−	15.8028i		−	0.893229i	0.894727	−	0.446614i	$-0.147370\pi$
$314$	−18.2074			−1.02750
$315$		0			0
$316$		−	9.74967i		−	0.548462i
$317$		−	11.5169i		−	0.646853i	−0.946253	−	0.323426i	$-0.895165\pi$
$317$		−	11.5169i		−	0.646853i	0.946253	−	0.323426i	$-0.104835\pi$
$318$		0			0
$319$			2.42064i			0.135530i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	−3.69334			−0.205822
$323$			12.7657i			0.710300i
$324$		0			0
$325$		−	4.74819i		−	0.263382i
$326$	−2.72596			−0.150977
$327$		0			0
$328$	−8.34118			−0.460565
$329$	−0.346540			−0.0191054
$330$		0			0
$331$		−	20.4890i		−	1.12618i	−0.826396	−	0.563090i	$-0.809613\pi$
$331$		−	20.4890i		−	1.12618i	0.826396	−	0.563090i	$-0.190387\pi$
$332$		−	15.7344i		−	0.863537i
$333$		0			0
$334$		−	1.15786i		−	0.0633555i
$335$	7.79027	−	2.51230i	0.425628	−	0.137261i
$336$		0			0
$337$			17.0388i			0.928160i	0.885793	+	0.464080i	$0.153615\pi$
$337$			17.0388i			0.928160i	−0.885793	+	0.464080i	$0.846385\pi$
$338$	−9.54531			−0.519197
$339$		0			0
$340$			4.08432i			0.221503i
$341$			0.719136i			0.0389434i
$342$		0			0
$343$		−	0.497263i		−	0.0268497i
$344$		−	1.30716i		−	0.0704773i
$345$		0			0
$346$		−	6.09577i		−	0.327711i
$347$	−9.75641			−0.523752			−0.261876	−	0.965102i	$-0.584341\pi$
$347$	−9.75641			−0.523752			−0.261876	+	0.965102i	$0.584341\pi$
$348$		0			0
$349$	−0.931651			−0.0498701			−0.0249351	−	0.999689i	$-0.507938\pi$
$349$	−0.931651			−0.0498701			−0.0249351	+	0.999689i	$0.507938\pi$
$350$			3.75929i			0.200943i
$351$		0			0
$352$	−0.476212			−0.0253822
$353$	6.33530			0.337194			0.168597	−	0.985685i	$-0.446076\pi$
$353$	6.33530			0.337194			0.168597	+	0.985685i	$0.446076\pi$
$354$		0			0
$355$			13.0140i			0.690712i
$356$		−	15.4861i		−	0.820764i
$357$		0			0
$358$	−4.62351			−0.244360
$359$		−	10.8400i		−	0.572114i	−0.958213	−	0.286057i	$-0.907655\pi$
$359$		−	10.8400i		−	0.572114i	0.958213	−	0.286057i	$-0.0923447\pi$
$360$		0			0
$361$	−9.23105			−0.485845
$362$	−2.93753			−0.154393
$363$		0			0
$364$	17.8498			0.935586
$365$	7.77609			0.407019
$366$		0			0
$367$		−	0.854953i		−	0.0446282i	−0.999751	−	0.0223141i	$-0.992897\pi$
$367$		−	0.854953i		−	0.0446282i	0.999751	−	0.0223141i	$-0.00710339\pi$
$368$			0.982457i			0.0512141i
$369$		0			0
$370$	4.80373			0.249734
$371$			5.32803i			0.276618i
$372$		0			0
$373$			27.6086i			1.42952i	0.699369	+	0.714760i	$0.253464\pi$
$373$			27.6086i			1.42952i	−0.699369	+	0.714760i	$0.746536\pi$
$374$			1.94500i			0.100574i
$375$		0			0
$376$			0.0921822i			0.00475393i
$377$	−24.1355			−1.24304
$378$		0			0
$379$			23.2433i			1.19393i	0.802268	+	0.596964i	$0.203627\pi$
$379$			23.2433i			1.19393i	−0.802268	+	0.596964i	$0.796373\pi$
$380$	3.12553			0.160336
$381$		0			0
$382$	15.7717			0.806952
$383$	−16.1316			−0.824285			−0.412143	−	0.911119i	$-0.635219\pi$
$383$	−16.1316			−0.824285			−0.412143	+	0.911119i	$0.635219\pi$
$384$		0			0
$385$			1.79022i			0.0912381i
$386$	−16.3858			−0.834017
$387$		0			0
$388$		−	6.02861i		−	0.306056i
$389$		−	0.469476i		−	0.0238034i	−0.999929	−	0.0119017i	$-0.996211\pi$
$389$		−	0.469476i		−	0.0238034i	0.999929	−	0.0119017i	$-0.00378852\pi$
$390$		0			0
$391$	4.01266			0.202929
$392$	−7.13228			−0.360234
$393$		0			0
$394$	22.0080			1.10875
$395$			9.74967i			0.490559i
$396$		0			0
$397$	−12.6232			−0.633542			−0.316771	−	0.948502i	$-0.602599\pi$
$397$	−12.6232			−0.633542			−0.316771	+	0.948502i	$0.602599\pi$
$398$	−23.6827			−1.18711
$399$		0			0
$400$	1.00000			0.0500000
$401$	31.8352			1.58978			0.794888	−	0.606756i	$-0.207530\pi$
$401$	31.8352			1.58978			0.794888	+	0.606756i	$0.207530\pi$
$402$		0			0
$403$	−7.17032			−0.357179
$404$	1.15912			0.0576685
$405$		0			0
$406$	19.1089			0.948357
$407$	2.28759			0.113392
$408$		0			0
$409$			12.4649i			0.616351i	0.951329	+	0.308176i	$0.0997185\pi$
$409$			12.4649i			0.616351i	−0.951329	+	0.308176i	$0.900282\pi$
$410$	8.34118			0.411941
$411$		0			0
$412$	16.4429			0.810083
$413$	−44.6535			−2.19726
$414$		0			0
$415$			15.7344i			0.772371i
$416$		−	4.74819i		−	0.232799i
$417$		0			0
$418$	1.48842			0.0728008
$419$		−	13.6786i		−	0.668243i	−0.942530	−	0.334121i	$-0.891560\pi$
$419$		−	13.6786i		−	0.668243i	0.942530	−	0.334121i	$-0.108440\pi$
$420$		0			0
$421$	−24.0100			−1.17018			−0.585088	−	0.810970i	$-0.698940\pi$
$421$	−24.0100			−1.17018			−0.585088	+	0.810970i	$0.698940\pi$
$422$	24.0556			1.17101
$423$		0			0
$424$	1.41730			0.0688300
$425$		−	4.08432i		−	0.198118i
$426$		0			0
$427$	−48.4072			−2.34259
$428$		−	11.5125i		−	0.556476i
$429$		0			0
$430$			1.30716i			0.0630369i
$431$			29.3250i			1.41254i	0.707944	+	0.706268i	$0.249623\pi$
$431$			29.3250i			1.41254i	−0.707944	+	0.706268i	$0.750377\pi$
$432$		0			0
$433$			30.9865i			1.48912i	0.667557	+	0.744559i	$0.267340\pi$
$433$			30.9865i			1.48912i	−0.667557	+	0.744559i	$0.732660\pi$
$434$	5.67697			0.272503
$435$		0			0
$436$			6.66875i			0.319375i
$437$		−	3.07070i		−	0.146892i
$438$		0			0
$439$	15.3340			0.731853			0.365927	−	0.930644i	$-0.380752\pi$
$439$	15.3340			0.731853			0.365927	+	0.930644i	$0.380752\pi$
$440$	0.476212			0.0227025
$441$		0			0
$442$	−19.3931			−0.922436
$443$	−12.0977			−0.574777			−0.287389	−	0.957814i	$-0.592787\pi$
$443$	−12.0977			−0.574777			−0.287389	+	0.957814i	$0.592787\pi$
$444$		0			0
$445$			15.4861i			0.734113i
$446$	4.78479			0.226567
$447$		0			0
$448$			3.75929i			0.177610i
$449$		−	10.4759i		−	0.494390i	−0.968966	−	0.247195i	$-0.920491\pi$
$449$		−	10.4759i		−	0.494390i	0.968966	−	0.247195i	$-0.0795088\pi$
$450$		0			0
$451$	3.97217			0.187042
$452$	11.9910			0.564009
$453$		0			0
$454$		−	2.84112i		−	0.133340i
$455$	−17.8498			−0.836813
$456$		0			0
$457$	−13.7599			−0.643660			−0.321830	−	0.946797i	$-0.604298\pi$
$457$	−13.7599			−0.643660			−0.321830	+	0.946797i	$0.604298\pi$
$458$		−	19.1224i		−	0.893531i
$459$		0			0
$460$		−	0.982457i		−	0.0458073i
$461$			9.57383i			0.445898i	0.974830	+	0.222949i	$0.0715683\pi$
$461$			9.57383i			0.445898i	−0.974830	+	0.222949i	$0.928432\pi$
$462$		0			0
$463$			16.7820i			0.779927i	0.920830	+	0.389963i	$0.127512\pi$
$463$			16.7820i			0.779927i	−0.920830	+	0.389963i	$0.872488\pi$
$464$		−	5.08310i		−	0.235977i
$465$		0			0
$466$	27.4779			1.27289
$467$			3.47247i			0.160687i	0.996767	+	0.0803434i	$0.0256017\pi$
$467$			3.47247i			0.160687i	−0.996767	+	0.0803434i	$0.974398\pi$
$468$		0			0
$469$	−9.44446	−	29.2859i	−0.436104	−	1.35230i
$470$		−	0.0921822i		−	0.00425205i
$471$		0			0
$472$			11.8782i			0.546738i
$473$			0.622485i			0.0286219i
$474$		0			0
$475$	−3.12553			−0.143409
$476$	15.3541			0.703756
$477$		0			0
$478$	−18.7228			−0.856360
$479$		−	41.3592i		−	1.88975i	−0.327432	−	0.944875i	$-0.606183\pi$
$479$		−	41.3592i		−	1.88975i	0.327432	−	0.944875i	$-0.393817\pi$
$480$		0			0
$481$			22.8090i			1.04000i
$482$	−1.66807			−0.0759786
$483$		0			0
$484$	−10.7732			−0.489692
$485$			6.02861i			0.273745i
$486$		0			0
$487$			19.7381i			0.894421i	0.894429	+	0.447210i	$0.147582\pi$
$487$			19.7381i			0.894421i	−0.894429	+	0.447210i	$0.852418\pi$
$488$			12.8767i			0.582899i
$489$		0			0
$490$	7.13228			0.322203
$491$			31.9919i			1.44377i	0.692012	+	0.721886i	$0.256725\pi$
$491$			31.9919i			1.44377i	−0.692012	+	0.721886i	$0.743275\pi$
$492$		0			0
$493$	−20.7610			−0.935029
$494$			14.8406i			0.667711i
$495$		0			0
$496$		−	1.51012i		−	0.0678062i
$497$	48.9235			2.19452
$498$		0			0
$499$			6.80024i			0.304420i	0.988348	+	0.152210i	$0.0486391\pi$
$499$			6.80024i			0.304420i	−0.988348	+	0.152210i	$0.951361\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	−1.33716			−0.0596803
$503$	−9.22761			−0.411439			−0.205719	−	0.978611i	$-0.565953\pi$
$503$	−9.22761			−0.411439			−0.205719	+	0.978611i	$0.565953\pi$
$504$		0			0
$505$	−1.15912			−0.0515802
$506$		−	0.467858i		−	0.0207988i
$507$		0			0
$508$	−1.66564			−0.0739008
$509$		−	27.0382i		−	1.19845i	−0.800582	−	0.599223i	$-0.795476\pi$
$509$		−	27.0382i		−	1.19845i	0.800582	−	0.599223i	$-0.204524\pi$
$510$		0			0
$511$		−	29.2326i		−	1.29317i
$512$	1.00000			0.0441942
$513$		0			0
$514$			21.0269i			0.927455i
$515$	−16.4429			−0.724560
$516$		0			0
$517$		−	0.0438983i		−	0.00193064i
$518$		−	18.0586i		−	0.793450i
$519$		0			0
$520$			4.74819i			0.208222i
$521$	−32.4507			−1.42169			−0.710846	−	0.703348i	$-0.751688\pi$
$521$	−32.4507			−1.42169			−0.710846	+	0.703348i	$0.751688\pi$
$522$		0			0
$523$	−28.1356			−1.23028			−0.615141	−	0.788417i	$-0.710901\pi$
$523$	−28.1356			−1.23028			−0.615141	+	0.788417i	$0.710901\pi$
$524$		−	19.5370i		−	0.853476i
$525$		0			0
$526$		−	12.8023i		−	0.558206i
$527$	−6.16779			−0.268673
$528$		0			0
$529$	22.0348			0.958034
$530$	−1.41730			−0.0615634
$531$		0			0
$532$		−	11.7498i		−	0.509418i
$533$			39.6055i			1.71551i
$534$		0			0
$535$			11.5125i			0.497728i
$536$	−7.79027	+	2.51230i	−0.336489	+	0.108515i
$537$		0			0
$538$			10.8084i			0.465983i
$539$	3.39648			0.146297
$540$		0			0
$541$		−	37.0985i		−	1.59499i	−0.603326	−	0.797495i	$-0.706158\pi$
$541$		−	37.0985i		−	1.59499i	0.603326	−	0.797495i	$-0.293842\pi$
$542$			8.07489i			0.346846i
$543$		0			0
$544$		−	4.08432i		−	0.175114i
$545$		−	6.66875i		−	0.285658i
$546$		0			0
$547$			42.6092i			1.82184i	0.412584	+	0.910919i	$0.364626\pi$
$547$			42.6092i			1.82184i	−0.412584	+	0.910919i	$0.635374\pi$
$548$	−9.32667			−0.398415
$549$		0			0
$550$	−0.476212			−0.0203058
$551$			15.8874i			0.676826i
$552$		0			0
$553$	36.6519			1.55860
$554$	−27.0641			−1.14984
$555$		0			0
$556$		−	19.2238i		−	0.815271i
$557$		−	7.15064i		−	0.302982i	−0.988459	−	0.151491i	$-0.951592\pi$
$557$		−	7.15064i		−	0.302982i	0.988459	−	0.151491i	$-0.0484075\pi$
$558$		0			0
$559$	−6.20664			−0.262513
$560$		−	3.75929i		−	0.158859i
$561$		0			0
$562$	−4.54839			−0.191862
$563$	−5.61751			−0.236750			−0.118375	−	0.992969i	$-0.537768\pi$
$563$	−5.61751			−0.236750			−0.118375	+	0.992969i	$0.537768\pi$
$564$		0			0
$565$	−11.9910			−0.504465
$566$	−6.53037			−0.274492
$567$		0			0
$568$		−	13.0140i		−	0.546056i
$569$		−	31.8233i		−	1.33410i	−0.745012	−	0.667051i	$-0.767556\pi$
$569$		−	31.8233i		−	1.33410i	0.745012	−	0.667051i	$-0.232444\pi$
$570$		0			0
$571$	27.2129			1.13883			0.569413	−	0.822052i	$-0.307171\pi$
$571$	27.2129			1.13883			0.569413	+	0.822052i	$0.307171\pi$
$572$			2.26115i			0.0945432i
$573$		0			0
$574$		−	31.3569i		−	1.30881i
$575$			0.982457i			0.0409713i
$576$		0			0
$577$			19.7471i			0.822082i	0.911617	+	0.411041i	$0.134835\pi$
$577$			19.7471i			0.822082i	−0.911617	+	0.411041i	$0.865165\pi$
$578$	0.318365			0.0132422
$579$		0			0
$580$			5.08310i			0.211064i
$581$	59.1502			2.45396
$582$		0			0
$583$	−0.674934			−0.0279529
$584$	−7.77609			−0.321777
$585$		0			0
$586$			20.0046i			0.826384i
$587$	1.12919			0.0466066			0.0233033	−	0.999728i	$-0.492582\pi$
$587$	1.12919			0.0466066			0.0233033	+	0.999728i	$0.492582\pi$
$588$		0			0
$589$			4.71992i			0.194481i
$590$		−	11.8782i		−	0.489017i
$591$		0			0
$592$	−4.80373			−0.197432
$593$	42.0678			1.72752			0.863759	−	0.503906i	$-0.168104\pi$
$593$	42.0678			1.72752			0.863759	+	0.503906i	$0.168104\pi$
$594$		0			0
$595$	−15.3541			−0.629458
$596$			7.15655i			0.293144i
$597$		0			0
$598$	4.66489			0.190762
$599$	48.6280			1.98689			0.993443	−	0.114329i	$-0.0364717\pi$
$599$	48.6280			1.98689			0.993443	+	0.114329i	$0.0364717\pi$
$600$		0			0
$601$	−13.9862			−0.570509			−0.285255	−	0.958452i	$-0.592078\pi$
$601$	−13.9862			−0.570509			−0.285255	+	0.958452i	$0.592078\pi$
$602$	4.91400			0.200280
$603$		0			0
$604$	12.9766			0.528010
$605$	10.7732			0.437994
$606$		0			0
$607$	20.5538			0.834251			0.417126	−	0.908849i	$-0.363038\pi$
$607$	20.5538			0.834251			0.417126	+	0.908849i	$0.363038\pi$
$608$	−3.12553			−0.126757
$609$		0			0
$610$		−	12.8767i		−	0.521361i
$611$	0.437699			0.0177074
$612$		0			0
$613$	−21.5799			−0.871603			−0.435801	−	0.900043i	$-0.643535\pi$
$613$	−21.5799			−0.871603			−0.435801	+	0.900043i	$0.643535\pi$
$614$	6.10978			0.246570
$615$		0			0
$616$		−	1.79022i		−	0.0721300i
$617$			1.00382i			0.0404121i	0.999796	+	0.0202061i	$0.00643223\pi$
$617$			1.00382i			0.0404121i	−0.999796	+	0.0202061i	$0.993568\pi$
$618$		0			0
$619$	37.9048			1.52352			0.761762	−	0.647857i	$-0.224335\pi$
$619$	37.9048			1.52352			0.761762	+	0.647857i	$0.224335\pi$
$620$			1.51012i			0.0606477i
$621$		0			0
$622$	−4.02204			−0.161269
$623$	58.2169			2.33241
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	15.8028i		−	0.631608i
$627$		0			0
$628$	−18.2074			−0.726552
$629$			19.6199i			0.782299i
$630$		0			0
$631$		−	1.39498i		−	0.0555331i	−0.999614	−	0.0277666i	$-0.991160\pi$
$631$		−	1.39498i		−	0.0555331i	0.999614	−	0.0277666i	$-0.00883951\pi$
$632$		−	9.74967i		−	0.387821i
$633$		0			0
$634$		−	11.5169i		−	0.457394i
$635$	1.66564			0.0660989
$636$		0			0
$637$			33.8654i			1.34180i
$638$			2.42064i			0.0958339i
$639$		0			0
$640$	−1.00000			−0.0395285
$641$	5.09665			0.201306			0.100653	−	0.994922i	$-0.467907\pi$
$641$	5.09665			0.201306			0.100653	+	0.994922i	$0.467907\pi$
$642$		0			0
$643$	−43.3752			−1.71055			−0.855275	−	0.518175i	$-0.826612\pi$
$643$	−43.3752			−1.71055			−0.855275	+	0.518175i	$0.826612\pi$
$644$	−3.69334			−0.145538
$645$		0			0
$646$			12.7657i			0.502258i
$647$	2.10582			0.0827884			0.0413942	−	0.999143i	$-0.486820\pi$
$647$	2.10582			0.0827884			0.0413942	+	0.999143i	$0.486820\pi$
$648$		0			0
$649$		−	5.65653i		−	0.222038i
$650$		−	4.74819i		−	0.186239i
$651$		0			0
$652$	−2.72596			−0.106757
$653$	−8.76287			−0.342918			−0.171459	−	0.985191i	$-0.554848\pi$
$653$	−8.76287			−0.342918			−0.171459	+	0.985191i	$0.554848\pi$
$654$		0			0
$655$			19.5370i			0.763372i
$656$	−8.34118			−0.325668
$657$		0			0
$658$	−0.346540			−0.0135095
$659$		−	1.90896i		−	0.0743625i	−0.999309	−	0.0371812i	$-0.988162\pi$
$659$		−	1.90896i		−	0.0743625i	0.999309	−	0.0371812i	$-0.0118379\pi$
$660$		0			0
$661$		−	1.65185i		−	0.0642496i	−0.999484	−	0.0321248i	$-0.989773\pi$
$661$		−	1.65185i		−	0.0642496i	0.999484	−	0.0321248i	$-0.0102274\pi$
$662$		−	20.4890i		−	0.796329i
$663$		0			0
$664$		−	15.7344i		−	0.610613i
$665$			11.7498i			0.455637i
$666$		0			0
$667$	4.99393			0.193366
$668$		−	1.15786i		−	0.0447991i
$669$		0			0
$670$	7.79027	−	2.51230i	0.300965	−	0.0970585i
$671$		−	6.13203i		−	0.236724i
$672$		0			0
$673$		−	6.90413i		−	0.266135i	−0.991107	−	0.133067i	$-0.957517\pi$
$673$		−	6.90413i		−	0.266135i	0.991107	−	0.133067i	$-0.0424827\pi$
$674$			17.0388i			0.656308i
$675$		0			0
$676$	−9.54531			−0.367127
$677$	5.02287			0.193045			0.0965224	−	0.995331i	$-0.469228\pi$
$677$	5.02287			0.193045			0.0965224	+	0.995331i	$0.469228\pi$
$678$		0			0
$679$	22.6633			0.869738
$680$			4.08432i			0.156626i
$681$		0			0
$682$			0.719136i			0.0275371i
$683$	−21.0049			−0.803730			−0.401865	−	0.915699i	$-0.631638\pi$
$683$	−21.0049			−0.803730			−0.401865	+	0.915699i	$0.631638\pi$
$684$		0			0
$685$	9.32667			0.356354
$686$		−	0.497263i		−	0.0189856i
$687$		0			0
$688$		−	1.30716i		−	0.0498350i
$689$		−	6.72960i		−	0.256377i
$690$		0			0
$691$	−37.8477			−1.43979			−0.719897	−	0.694080i	$-0.755811\pi$
$691$	−37.8477			−1.43979			−0.719897	+	0.694080i	$0.755811\pi$
$692$		−	6.09577i		−	0.231726i
$693$		0			0
$694$	−9.75641			−0.370348
$695$			19.2238i			0.729200i
$696$		0			0
$697$			34.0680i			1.29042i
$698$	−0.931651			−0.0352635
$699$		0			0
$700$			3.75929i			0.142088i
$701$	−15.8919			−0.600228			−0.300114	−	0.953903i	$-0.597025\pi$
$701$	−15.8919			−0.600228			−0.300114	+	0.953903i	$0.597025\pi$
$702$		0			0
$703$	15.0142			0.566272
$704$	−0.476212			−0.0179479
$705$		0			0
$706$	6.33530			0.238432
$707$			4.35748i			0.163880i
$708$		0			0
$709$	−49.2435			−1.84938			−0.924689	−	0.380723i	$-0.875675\pi$
$709$	−49.2435			−1.84938			−0.924689	+	0.380723i	$0.875675\pi$
$710$			13.0140i			0.488407i
$711$		0			0
$712$		−	15.4861i		−	0.580368i
$713$	1.48362			0.0555622
$714$		0			0
$715$		−	2.26115i		−	0.0845621i
$716$	−4.62351			−0.172789
$717$		0			0
$718$		−	10.8400i		−	0.404546i
$719$		−	6.40038i		−	0.238694i	−0.992853	−	0.119347i	$-0.961920\pi$
$719$		−	6.40038i		−	0.238694i	0.992853	−	0.119347i	$-0.0380801\pi$
$720$		0			0
$721$			61.8136i			2.30206i
$722$	−9.23105			−0.343544
$723$		0			0
$724$	−2.93753			−0.109172
$725$		−	5.08310i		−	0.188782i
$726$		0			0
$727$			22.1941i			0.823132i	0.911380	+	0.411566i	$0.135018\pi$
$727$			22.1941i			0.823132i	−0.911380	+	0.411566i	$0.864982\pi$
$728$	17.8498			0.661559
$729$		0			0
$730$	7.77609			0.287806
$731$	−5.33885			−0.197465
$732$		0			0
$733$		−	23.0141i		−	0.850045i	−0.905183	−	0.425022i	$-0.860266\pi$
$733$		−	23.0141i		−	0.850045i	0.905183	−	0.425022i	$-0.139734\pi$
$734$		−	0.854953i		−	0.0315569i
$735$		0			0
$736$			0.982457i			0.0362138i
$737$	3.70982	−	1.19639i	0.136653	−	0.0440694i
$738$		0			0
$739$		−	25.9654i		−	0.955152i	−0.878590	−	0.477576i	$-0.841516\pi$
$739$		−	25.9654i		−	0.955152i	0.878590	−	0.477576i	$-0.158484\pi$
$740$	4.80373			0.176589
$741$		0			0
$742$			5.32803i			0.195598i
$743$			11.9643i			0.438928i	0.975621	+	0.219464i	$0.0704308\pi$
$743$			11.9643i			0.438928i	−0.975621	+	0.219464i	$0.929569\pi$
$744$		0			0
$745$		−	7.15655i		−	0.262196i
$746$			27.6086i			1.01082i
$747$		0			0
$748$			1.94500i			0.0711163i
$749$	43.2787			1.58137
$750$		0			0
$751$	−17.1205			−0.624738			−0.312369	−	0.949961i	$-0.601122\pi$
$751$	−17.1205			−0.624738			−0.312369	+	0.949961i	$0.601122\pi$
$752$			0.0921822i			0.00336154i
$753$		0			0
$754$	−24.1355			−0.878965
$755$	−12.9766			−0.472267
$756$		0			0
$757$		−	26.9992i		−	0.981301i	−0.871356	−	0.490651i	$-0.836759\pi$
$757$		−	26.9992i		−	0.981301i	0.871356	−	0.490651i	$-0.163241\pi$
$758$			23.2433i			0.844235i
$759$		0			0
$760$	3.12553			0.113375
$761$			25.4688i			0.923243i	0.887077	+	0.461622i	$0.152732\pi$
$761$			25.4688i			0.923243i	−0.887077	+	0.461622i	$0.847268\pi$
$762$		0			0
$763$	−25.0698			−0.907587
$764$	15.7717			0.570601
$765$		0			0
$766$	−16.1316			−0.582858
$767$	56.3999			2.03648
$768$		0			0
$769$		−	15.9169i		−	0.573978i	−0.957934	−	0.286989i	$-0.907346\pi$
$769$		−	15.9169i		−	0.573978i	0.957934	−	0.286989i	$-0.0926543\pi$
$770$			1.79022i			0.0645151i
$771$		0			0
$772$	−16.3858			−0.589739
$773$		−	53.2607i		−	1.91565i	−0.287346	−	0.957827i	$-0.592773\pi$
$773$		−	53.2607i		−	1.91565i	0.287346	−	0.957827i	$-0.407227\pi$
$774$		0			0
$775$		−	1.51012i		−	0.0542450i
$776$		−	6.02861i		−	0.216414i
$777$		0			0
$778$		−	0.469476i		−	0.0168315i
$779$	26.0706			0.934077
$780$		0			0
$781$			6.19743i			0.221761i
$782$	4.01266			0.143493
$783$		0			0
$784$	−7.13228			−0.254724
$785$	18.2074			0.649848
$786$		0			0
$787$			13.1167i			0.467559i	0.972290	+	0.233779i	$0.0751093\pi$
$787$			13.1167i			0.467559i	−0.972290	+	0.233779i	$0.924891\pi$
$788$	22.0080			0.784004
$789$		0			0
$790$			9.74967i			0.346878i
$791$			45.0777i			1.60278i
$792$		0			0
$793$	61.1409			2.17118
$794$	−12.6232			−0.447982
$795$		0			0
$796$	−23.6827			−0.839411
$797$		−	7.72777i		−	0.273732i	−0.990590	−	0.136866i	$-0.956297\pi$
$797$		−	7.72777i		−	0.273732i	0.990590	−	0.136866i	$-0.0437029\pi$
$798$		0			0
$799$	0.376501			0.0133197
$800$	1.00000			0.0353553
$801$		0			0
$802$	31.8352			1.12414
$803$	3.70307			0.130679
$804$		0			0
$805$	3.69334			0.130173
$806$	−7.17032			−0.252564
$807$		0			0
$808$	1.15912			0.0407778
$809$	5.10824			0.179596			0.0897981	−	0.995960i	$-0.471378\pi$
$809$	5.10824			0.179596			0.0897981	+	0.995960i	$0.471378\pi$
$810$		0			0
$811$			8.29450i			0.291259i	0.989339	+	0.145630i	$0.0465208\pi$
$811$			8.29450i			0.291259i	−0.989339	+	0.145630i	$0.953479\pi$
$812$	19.1089			0.670590
$813$		0			0
$814$	2.28759			0.0801801
$815$	2.72596			0.0954861
$816$		0			0
$817$			4.08557i			0.142936i
$818$			12.4649i			0.435826i
$819$		0			0
$820$	8.34118			0.291287
$821$			37.7331i			1.31690i	0.752626	+	0.658448i	$0.228787\pi$
$821$			37.7331i			1.31690i	−0.752626	+	0.658448i	$0.771213\pi$
$822$		0			0
$823$	45.3410			1.58049			0.790244	−	0.612793i	$-0.209954\pi$
$823$	45.3410			1.58049			0.790244	+	0.612793i	$0.209954\pi$
$824$	16.4429			0.572815
$825$		0			0
$826$	−44.6535			−1.55370
$827$			30.4750i			1.05972i	0.848086	+	0.529859i	$0.177755\pi$
$827$			30.4750i			1.05972i	−0.848086	+	0.529859i	$0.822245\pi$
$828$		0			0
$829$	−19.3701			−0.672753			−0.336376	−	0.941728i	$-0.609201\pi$
$829$	−19.3701			−0.672753			−0.336376	+	0.941728i	$0.609201\pi$
$830$			15.7344i			0.546149i
$831$		0			0
$832$		−	4.74819i		−	0.164614i
$833$			29.1305i			1.00931i
$834$		0			0
$835$			1.15786i			0.0400695i
$836$	1.48842			0.0514779
$837$		0			0
$838$		−	13.6786i		−	0.472519i
$839$			21.7574i			0.751150i	0.926792	+	0.375575i	$0.122555\pi$
$839$			21.7574i			0.751150i	−0.926792	+	0.375575i	$0.877445\pi$
$840$		0			0
$841$	3.16205			0.109036
$842$	−24.0100			−0.827439
$843$		0			0
$844$	24.0556			0.828026
$845$	9.54531			0.328369
$846$		0			0
$847$		−	40.4997i		−	1.39159i
$848$	1.41730			0.0486702
$849$		0			0
$850$		−	4.08432i		−	0.140091i
$851$		−	4.71946i		−	0.161781i
$852$		0			0
$853$	39.5244			1.35329			0.676644	−	0.736310i	$-0.263434\pi$
$853$	39.5244			1.35329			0.676644	+	0.736310i	$0.263434\pi$
$854$	−48.4072			−1.65646
$855$		0			0
$856$		−	11.5125i		−	0.393488i
$857$	41.2586			1.40937			0.704683	−	0.709523i	$-0.251089\pi$
$857$	41.2586			1.40937			0.704683	+	0.709523i	$0.251089\pi$
$858$		0			0
$859$	8.56988			0.292401			0.146200	−	0.989255i	$-0.453296\pi$
$859$	8.56988			0.292401			0.146200	+	0.989255i	$0.453296\pi$
$860$			1.30716i			0.0445738i
$861$		0			0
$862$			29.3250i			0.998814i
$863$		−	13.3930i		−	0.455903i	−0.973673	−	0.227951i	$-0.926797\pi$
$863$		−	13.3930i		−	0.455903i	0.973673	−	0.227951i	$-0.0732027\pi$
$864$		0			0
$865$			6.09577i			0.207262i
$866$			30.9865i			1.05297i
$867$		0			0
$868$	5.67697			0.192689
$869$			4.64291i			0.157500i
$870$		0			0
$871$	11.9289	+	36.9897i	0.404194	+	1.25335i
$872$			6.66875i			0.225832i
$873$		0			0
$874$		−	3.07070i		−	0.103868i
$875$		−	3.75929i		−	0.127087i
$876$		0			0
$877$	37.6541			1.27149			0.635744	−	0.771900i	$-0.280693\pi$
$877$	37.6541			1.27149			0.635744	+	0.771900i	$0.280693\pi$
$878$	15.3340			0.517498
$879$		0			0
$880$	0.476212			0.0160531
$881$		−	27.1033i		−	0.913135i	−0.889689	−	0.456567i	$-0.849079\pi$
$881$		−	27.1033i		−	0.913135i	0.889689	−	0.456567i	$-0.150921\pi$
$882$		0			0
$883$		−	47.1167i		−	1.58560i	−0.609480	−	0.792802i	$-0.708622\pi$
$883$		−	47.1167i		−	1.58560i	0.609480	−	0.792802i	$-0.291378\pi$
$884$	−19.3931			−0.652261
$885$		0			0
$886$	−12.0977			−0.406429
$887$		−	9.07303i		−	0.304642i	−0.988331	−	0.152321i	$-0.951325\pi$
$887$		−	9.07303i		−	0.304642i	0.988331	−	0.152321i	$-0.0486748\pi$
$888$		0			0
$889$		−	6.26162i		−	0.210008i
$890$			15.4861i			0.519097i
$891$		0			0
$892$	4.78479			0.160207
$893$		−	0.288118i		−	0.00964151i
$894$		0			0
$895$	4.62351			0.154547
$896$			3.75929i			0.125589i
$897$		0			0
$898$		−	10.4759i		−	0.349586i
$899$	−7.67608			−0.256012
$900$		0			0
$901$		−	5.78869i		−	0.192849i
$902$	3.97217			0.132259
$903$		0			0
$904$	11.9910			0.398815
$905$	2.93753			0.0976468
$906$		0			0
$907$	−20.7096			−0.687650			−0.343825	−	0.939034i	$-0.611723\pi$
$907$	−20.7096			−0.687650			−0.343825	+	0.939034i	$0.611723\pi$
$908$		−	2.84112i		−	0.0942859i
$909$		0			0
$910$	−17.8498			−0.591716
$911$			5.60058i			0.185555i	0.995687	+	0.0927777i	$0.0295746\pi$
$911$			5.60058i			0.185555i	−0.995687	+	0.0927777i	$0.970425\pi$
$912$		0			0
$913$			7.49291i			0.247979i
$914$	−13.7599			−0.455136
$915$		0			0
$916$		−	19.1224i		−	0.631821i
$917$	73.4451			2.42537
$918$		0			0
$919$		−	22.5215i		−	0.742916i	−0.928450	−	0.371458i	$-0.878858\pi$
$919$		−	22.5215i		−	0.742916i	0.928450	−	0.371458i	$-0.121142\pi$
$920$		−	0.982457i		−	0.0323906i
$921$		0			0
$922$			9.57383i			0.315297i
$923$	−61.7930			−2.03394
$924$		0			0
$925$	−4.80373			−0.157946
$926$			16.7820i			0.551492i
$927$		0			0
$928$		−	5.08310i		−	0.166861i
$929$	−40.6612			−1.33405			−0.667026	−	0.745034i	$-0.732433\pi$
$929$	−40.6612			−1.33405			−0.667026	+	0.745034i	$0.732433\pi$
$930$		0			0
$931$	22.2922			0.730596
$932$	27.4779			0.900069
$933$		0			0
$934$			3.47247i			0.113623i
$935$		−	1.94500i		−	0.0636083i
$936$		0			0
$937$			41.2617i			1.34796i	0.738749	+	0.673981i	$0.235417\pi$
$937$			41.2617i			1.34796i	−0.738749	+	0.673981i	$0.764583\pi$
$938$	−9.44446	−	29.2859i	−0.308372	−	0.956219i
$939$		0			0
$940$		−	0.0921822i		−	0.00300665i
$941$	−56.0574			−1.82742			−0.913710	−	0.406366i	$-0.866796\pi$
$941$	−56.0574			−1.82742			−0.913710	+	0.406366i	$0.866796\pi$
$942$		0			0
$943$		−	8.19485i		−	0.266861i
$944$			11.8782i			0.386602i
$945$		0			0
$946$			0.622485i			0.0202387i
$947$		−	4.93853i		−	0.160481i	−0.996776	−	0.0802403i	$-0.974431\pi$
$947$		−	4.93853i		−	0.160481i	0.996776	−	0.0802403i	$-0.0255688\pi$
$948$		0			0
$949$			36.9224i			1.19855i
$950$	−3.12553			−0.101406
$951$		0			0
$952$	15.3541			0.497630
$953$		−	33.4644i		−	1.08402i	−0.840372	−	0.542010i	$-0.817664\pi$
$953$		−	33.4644i		−	1.08402i	0.840372	−	0.542010i	$-0.182336\pi$
$954$		0			0
$955$	−15.7717			−0.510361
$956$	−18.7228			−0.605538
$957$		0			0
$958$		−	41.3592i		−	1.33625i
$959$		−	35.0617i		−	1.13220i
$960$		0			0
$961$	28.7195			0.926437
$962$			22.8090i			0.735392i
$963$		0			0
$964$	−1.66807			−0.0537250
$965$	16.3858			0.527478
$966$		0			0
$967$	26.9534			0.866763			0.433381	−	0.901211i	$-0.357320\pi$
$967$	26.9534			0.866763			0.433381	+	0.901211i	$0.357320\pi$
$968$	−10.7732			−0.346264
$969$		0			0
$970$			6.02861i			0.193567i
$971$		−	18.1355i		−	0.581995i	−0.956724	−	0.290998i	$-0.906013\pi$
$971$		−	18.1355i		−	0.581995i	0.956724	−	0.290998i	$-0.0939872\pi$
$972$		0			0
$973$	72.2679			2.31680
$974$			19.7381i			0.632451i
$975$		0			0
$976$			12.8767i			0.412172i
$977$			6.26366i			0.200392i	0.994968	+	0.100196i	$0.0319470\pi$
$977$			6.26366i			0.200392i	−0.994968	+	0.100196i	$0.968053\pi$
$978$		0			0
$979$			7.37469i			0.235696i
$980$	7.13228			0.227832
$981$		0			0
$982$			31.9919i			1.02090i
$983$	−45.7854			−1.46033			−0.730163	−	0.683273i	$-0.760556\pi$
$983$	−45.7854			−1.46033			−0.730163	+	0.683273i	$0.760556\pi$
$984$		0			0
$985$	−22.0080			−0.701235
$986$	−20.7610			−0.661165
$987$		0			0
$988$			14.8406i			0.472143i
$989$	1.28423			0.0408361
$990$		0			0
$991$			23.5086i			0.746776i	0.927675	+	0.373388i	$0.121804\pi$
$991$			23.5086i			0.746776i	−0.927675	+	0.373388i	$0.878196\pi$
$992$		−	1.51012i		−	0.0479462i
$993$		0			0
$994$	48.9235			1.55176
$995$	23.6827			0.750792
$996$		0			0
$997$	22.1693			0.702110			0.351055	−	0.936355i	$-0.385823\pi$
$997$	22.1693			0.702110			0.351055	+	0.936355i	$0.385823\pi$
$998$			6.80024i			0.215258i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.j.2411.13	yes	16
3.2	odd	2		6030.2.d.i.2411.13	yes	16
67.66	odd	2		6030.2.d.i.2411.4	✓	16
201.200	even	2	inner	6030.2.d.j.2411.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.i.2411.4	✓	16	67.66	odd	2
6030.2.d.i.2411.13	yes	16	3.2	odd	2
6030.2.d.j.2411.4	yes	16	201.200	even	2	inner
6030.2.d.j.2411.13	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.75929i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.75929i q^{7} +1.00000 q^{8} -1.00000 q^{10} -0.476212 q^{11} -4.74819i q^{13} +3.75929i q^{14} +1.00000 q^{16} -4.08432i q^{17} -3.12553 q^{19} -1.00000 q^{20} -0.476212 q^{22} +0.982457i q^{23} +1.00000 q^{25} -4.74819i q^{26} +3.75929i q^{28} -5.08310i q^{29} -1.51012i q^{31} +1.00000 q^{32} -4.08432i q^{34} -3.75929i q^{35} -4.80373 q^{37} -3.12553 q^{38} -1.00000 q^{40} -8.34118 q^{41} -1.30716i q^{43} -0.476212 q^{44} +0.982457i q^{46} +0.0921822i q^{47} -7.13228 q^{49} +1.00000 q^{50} -4.74819i q^{52} +1.41730 q^{53} +0.476212 q^{55} +3.75929i q^{56} -5.08310i q^{58} +11.8782i q^{59} +12.8767i q^{61} -1.51012i q^{62} +1.00000 q^{64} +4.74819i q^{65} +(-7.79027 + 2.51230i) q^{67} -4.08432i q^{68} -3.75929i q^{70} -13.0140i q^{71} -7.77609 q^{73} -4.80373 q^{74} -3.12553 q^{76} -1.79022i q^{77} -9.74967i q^{79} -1.00000 q^{80} -8.34118 q^{82} -15.7344i q^{83} +4.08432i q^{85} -1.30716i q^{86} -0.476212 q^{88} -15.4861i q^{89} +17.8498 q^{91} +0.982457i q^{92} +0.0921822i q^{94} +3.12553 q^{95} -6.02861i q^{97} -7.13228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} - 16 q^{5} + 16 q^{8} - 16 q^{10} - 20 q^{11} + 16 q^{16} - 8 q^{19} - 16 q^{20} - 20 q^{22} + 16 q^{25} + 16 q^{32} + 32 q^{37} - 8 q^{38} - 16 q^{40} - 8 q^{41} - 20 q^{44} - 88 q^{49} + 16 q^{50} + 8 q^{53} + 20 q^{55} + 16 q^{64} + 4 q^{67} + 16 q^{73} + 32 q^{74} - 8 q^{76} - 16 q^{80} - 8 q^{82} - 20 q^{88} + 40 q^{91} + 8 q^{95} - 88 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 - 16 * q^5 + 16 * q^8 - 16 * q^10 - 20 * q^11 + 16 * q^16 - 8 * q^19 - 16 * q^20 - 20 * q^22 + 16 * q^25 + 16 * q^32 + 32 * q^37 - 8 * q^38 - 16 * q^40 - 8 * q^41 - 20 * q^44 - 88 * q^49 + 16 * q^50 + 8 * q^53 + 20 * q^55 + 16 * q^64 + 4 * q^67 + 16 * q^73 + 32 * q^74 - 8 * q^76 - 16 * q^80 - 8 * q^82 - 20 * q^88 + 40 * q^91 + 8 * q^95 - 88 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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